Analysis and Suppression of Thrust Fluctuations in Halbach-type Permanent Magnet Synchronous Linear Motors

To address the thrust fluctuation problem of a bilateral type permanent magnet synchronous linear motor (DSPMSLM) with a Halbach permanent magnet array, a DSPMLSM with Halbach permanent magnet array structure is designed using ANSYS software. And based on this motor, the thrust fluctuations of the DSPMLSM are suppressed by using methods such as changing the end tooth height of the armature core, the tooth slot structure, and the opening of an auxiliary slot in the primary yoke. Firstly, the causes of thrust fluctuations in these areas were analyzed. Secondly, MAXWELL simulations were used to demonstrate that the selection of a suitable end tooth height could suppress end forces and that the use of a suitable-sized half-open slot and an auxiliary slot in the primary yoke could effectively reduce slot forces. All three methods can be used to suppress motor thrust fluctuations, reducing the thrust fluctuations of the DSPMLSM from 11.4% to 3.28% and increasing the average output thrust, which provides a reference for further optimization of the output performance of the linear motor.


Introduction
In recent years, with the continuous increase in building height and the rapid development of rail transport, higher requirements have been placed on the various properties of lifting and transmission devices.The permanent magnet synchronous linear motor has the respective characteristics of both linear motors and permanent magnet motors and is capable of linear motion using electrical energy, with a simple drive structure, good dynamic characteristics, and high thrust density, making it suitable for high-speed, high-precision linear drive applications.The main cause of thrust fluctuations in linear motors lies in their own end effect and cogging effect, and the end effect and cogging effect corresponding to the resulting end force and cogging force is the main source of fluctuations, eventually leading to noise and vibration, affecting the life of the motor.So, the suppression of thrust fluctuations is a decisive factor in the output performance of linear motors [1][2][3][4].
Since the introduction of Halbach permanent magnet arrays in 1979, scholars at home and abroad have conducted a lot of research on the application of Halbach arrays to the field of linear motors.Halbach arrays are able to enhance the unilateral magnetic field and the magnetic field variation is more in line with the sinusoidal law.In [5], linear motors with Halbach permanent magnet arrays were compared with conventional permanent magnet arrays to show an improvement in output thrust, thrust fluctuation, and start-up response time performance.In [6], the optimal pole arc factor and pole chamfering method were used to weaken the slot normal force, and the optimized core length and end tooth shape were used to weaken the end normal force in order to achieve a combination of weakening PMLM normal force fluctuations.In [7], the end effect, slot effect, and thrust ripple were identified as key factors contributing to thrust fluctuations in the output of PMM synchronous linear motors, and measures were discussed and analyzed to reduce the fluctuations.In [8], a simulation analysis of various slot types of linear motors was carried out to determine the different effects of different primary slot types on motor output thrust and harmonic generation.In [9], the output performance of the motor was improved by optimizing the width of the main permanent magnets of the Halbach array.
In this paper, the characteristics of the Halbach permanent magnet array and bilateral type permanent magnet linear motor are combined, and ANSYS software is used to simulate the bilateral type permanent magnet synchronous linear motor (DSPMSLM) of this Halbach permanent magnet array structure.The thrust fluctuation problem of the Halbach array structure of the bilateral type PM synchronous linear is firstly analyzed in theoretical terms to determine the causes of end force and magnetic resistance, and the key factors affecting the magnitude of end force and magnetic resistance.The feasibility of these three methods is verified by MAXWELL simulation.

Principle of operation of permanent magnet synchronous linear motors
A linear motor is a transmission device that converts electrical energy directly into mechanical energy for linear motion without any intermediate conversion mechanism.Structurally, a linear motor can be seen as a conversion from a permanent magnet rotary motor, as shown in Figure 1.
Figure 1.The process of changing from a rotating motor to a linear motor When three symmetrical currents are passed through the windings of a rotating motor, a rotating magnetic field is generated in the air gap located between the stator and rotor at a synchronous speed.The speed of the rotating motor air gap magnetic field is In the equation, f is the frequency of the three symmetrical currents passing through the winding, and p is the number of pole pairs.The linear motor is not only structurally equivalent to a rotary motor but also operates on a similar principle to a rotary motor.The difference is that the magnetic field generated in a linear motor is a traveling wave field in translational motion, rather than rotating.Similarly, it can be deduced that the traveling magnetic field of a linear motor moves at a speed of where W is the polar distance and v unit is m/s.
Slotting and end effects are the main causes of magnetic resistance in linear motors [10].Ideally, the equation for the magnetic resistance per unit volume from [11] where 0 O is a constant, n O is the nth harmonic amplitude, s W is the slot pitch, m f is the amplitude of the mth harmonic, W is the polar distance, p is the polar logarithm, and det F is the magnetic resistance.

DSPLSM finite element simulation model
The DSPLSM in question is a bilateral 12-slot/11-pole permanent magnet synchronous linear motor, using alternating tooth windings.The primary consists of a winding and an armature core, where fractional slots are used to increase thrust and reduce thrust fluctuations, and a bilateral secondary motor structure is used to weaken normal forces.The secondary consists of permanent magnets and a back iron, with Halbach permanent magnets fixed to the secondary back iron.The motor parameters are shown in Table 1, and the motor model is shown in Figure 2.  In this paper, ANSYS software is used to establish the simulation model DSPMLSM.The primary iron core material is DW315-50, the secondary material is steel-1010, the winding is copper, and the permanent magnet material is N35 series.

End tooth height optimization
Unlike rotating motors, the end effect is only present in linear motors and is caused by the distortion of the magnetic field at the primary core at the side end of the linear motor.Unlike the unilateral type, the end forces generated by the end effect are one of the main causes of thrust fluctuations in linear motors as the normal forces of the bilateral type permanent magnet linear synchronous motor cancel up and down to very small values, so optimization of the end tooth heights can reduce the end forces to a large extent and thus improve the thrust fluctuations in linear motors.Figure 3 shows an illustration of the optimized height of the end teeth.4 and Figure 5. From Figure 4 and Figure 5, it can be seen that when the length of the end tooth is reduced to 31 mm, one side is reduced by 5 mm and the bilateral side is reduced by a total of 10 mm.The peak value of the thrust fluctuation at this time is 71.9 N and the thrust fluctuation at this time is approximately 3.91%.Compared to the original model end tooth length of 36 mm, it can be seen that the magnetic resistance has decreased from 209.4 N to 71.9 N and the percentage of thrust fluctuation has decreased by approximately 65.7%.

Optimization of the notch structure
Similar to the slot torque of a rotary motor, another important cause of linear thrust fluctuations is the slot force, which is essentially the result of the armature slotting, causing the magnetic flux between the permanent magnet pole and the armature slot to vary at different positions during the movement of the primary core, which in turn causes the magnetic field energy storage to change, resulting in a slot force.
The slot structure is one of the factors influencing the magnitude of the slot force on the armature core.The full-open slot of the original design can be replaced by a half-open slot, and the slot size can be optimized to reduce the slot force.Secondly, the slotting of the primary yoke is also one of the factors influencing the amount of slotting force on the armature core, different primary yoke slotting sizes and slotting shapes have different effects on the slotting force.Figure 6 and Figure 7 show the average values of peak-to-peak and output thrust for different rectangular auxiliary slots.It is known that when the rectangular slot size is 0.6*0.2(mm*mm), the thrust fluctuation value is minimum.

Thrust fluctuation result suppression analysis
The simulation results for the three simulation models are shown in Figure 8 and Figure 9, where Force 1 is the waveform with a semi-open slot and the primary tooth yoke slotted, Force 2 is the waveform with the end tooth height optimized and Force 3 is the waveform of the original model.
From Figure 8 and Figure 9, it can be seen that when the end tooth length is 31 mm, the average output thrust is 1830.1 N and the peak value of the thrust fluctuation is 71.9 N, which is about 3.

Conclusion
In this paper, a prototype model of DSPMLSM with a Halbach permanent magnet array structure is designed using ANSYS software, and its electromagnetic characteristics are simulated and analyzed.
On the basis of this paper, to address the shortcomings of the DSPMLSM with Halbach permanent magnet array structure in terms of large thrust fluctuations, a method to suppress thrust fluctuations by adopting suitable end tooth height at the end of the armature core, tooth slot structure and auxiliary slot in the primary tooth yoke is proposed, and the following conclusions are obtained: 1) Optimization of the end tooth height can significantly suppress the magnetic field distortion due to open ends of the magnetic circuit in linear motors, thereby weakening the thrust fluctuations caused by end forces.
2) Compared to a full slot, a half slot in a Halbach-type DSPMLSM can reduce output fluctuations and at the same time increase the average thrust of the motor output.And a combination of a half slot and an auxiliary slot can further suppress output fluctuations.

Figure 2 .
Figure 2. DSPMLSM simulation model of Halbach permanent magnet array structure

Figure 3 .
Figure 3. Schematic diagram of end tooth height optimization

Figure 4 .
Figure 4.The peak value of thrust Figure 5.Thrust force force at different end tooth heights at different end tooth heights

Figure 6 .
Figure 6.Output thrust for Figure 7.The peak value of output thrust different rectangular auxiliary slots for different rectangular auxiliary slots 91%.The percentage reduction is approximately 65.7%.When the Halbach permanent magnet array structure of the bilateral permanent magnet synchronous linear motor is a fully open slot, the average output thrust is 1830.1 N, the peak thrust fluctuation is 71.9 N and the thrust fluctuation is about 3.91%, while the linear motor is a half-open slot, the average output thrust is 1922.3N, the peak thrust fluctuation is 63.1 N and the thrust fluctuations of about 3.28%.The percentage of thrust fluctuations decreases by about 19.2%, and the linear motor output thrust slightly increases.

Figure 8 .
Figure 8.Output waveform comparison chart Figure 9. Magnetic resistance comparison chart